The range rule of thumb tells us that the range of the data is approximately equal to 4 times the standard deviation. This rule is the most accurate for data that has a normal distribution and when the sample size is around 30.
As we know from the empirical rule, 95% of the data values in a normal distribution lie within 4 standard deviations. Hence the rule is fairly accurate for normal distribution.
Example:
Consider the following set of data:
1, 3, 4, 6, 7, 7, 8, 12
Approximate the range using the range rule of thumb.
Solution: The true value of the range is 12-1 = 11.
We calculate the standard deviation using the formula,
σ = √[∑(xi-µ)2/n] = 3.16
So Range = 4*σ = 4*3.16 = 12.48 approximately, which is close to the true value of the range.
Improvements on the Range Rule:
The range rule states that,
Range = 4*σ where σ is the standard deviation of the data.
We can improve upon the range rule by replacing the factor 4 with a(n) where a(n) is a function dependent on the sample size. The range rule then becomes,
Range = a(n)*σ
We can find suitable functions for a(n) depending on the population distribution by running simulations. As seen before a(30) = 4 for normal distributions.
Using the Range Rule to Find Outliers:
We can use the range rule to find outliers in our given data set. We can do this by adding and subtracting two standard deviations from our mean.
Any data value lying beyond these bounds would be considered as beyond the acceptable range and become outliers.
